Math 311-102 (Summer 2006) Name 1 Test II Instructions: Show all work in your bluebook. Cell phones, laptops, calculators that do linear algebra or calculus, and other such devices are not allowed. 1. (10 pts.) Determine whether or not S = {f ∈ C[2, 5] | f (2) = f (5)} is a subspace of C[2, 5]. 2. (5 pts.) For the subsets of P3 below, which cannot be linearly independent? Which cannot span? Briefly explain how you are getting your answers. (a) {1 − 2x, x2 , x + x3 } (b) {x2 , x3 , 1 − 3x + 10x2 − 5x3 } (c) {1, x2 − 2, x2 − 6} (d) {x − 3x2 , x3 , x2 − 3x + 2, x − 1, 5x2 − 7} (e) {1, x, x3 , x − x2 , 3x − 1, 2x3 }. 3. (10 pts.) Let B = {1, ex , e2x }. Show that B is LI and find the coordinates of f (x) = (1 − 3ex )2 . 4. (15 pts.) Find bases for the column space, null space, and row space of C, and state the rank and nullity of C. What should these sum to? Do they? 1 −2 −1 −3 2 4 C = −1 2 2 −4 −3 −7 5. Let L : P2 → P2 be defined by L[p] = (x2 − 1)p00 + (x + 3)p0 − 4p . (a) (5 pts.) Show that L is linear. (b) (5 pts.) Find the matrix A of L relative to the basis B = {1, x, x2 }. (c) (5 pts.) Find a basis for the null space of L. What is the rank of L? 6. (15 pts.) Consider the matrix A below. Find the eigenvalues and eigenvectors of A. Also, find a diagonal matrix Λ and an invertible matrix S for which A = SΛS −1 . 2 0 1 A= 0 3 0 . 1 0 2 Rπ 7. (15 pts.) Consider the inner product hf, gi = 0 f (x)g(x)dx on continuous functions C[0, π]. Use the Gram-Schmidt process to turn 2 {1, cos(x), (x)} into an orthogonal set relative this inner product. R ton−2 R cos n−1 1 n−1 n (Hint: cos (x)dx = n cos (x) sin(x) + n cos (x)dx.) R1 8. Consider the inner product hf, gi = −1 f (x)g(x)dx on continuous functions C[−1, 1]. (a) (5 pts.) Find the Gram matrix A for {1, x, x2 }. (b) (10 pts.) Use A and the given inner product to find the best quadratic fit to f (x) = |x|. 2